| Sample ID | x_bar | s |
|---|---|---|
| 1 | -0.047 | 0.981 |
| 2 | 0.178 | 0.835 |
| 3 | 0.024 | 0.870 |
| 4 | -0.094 | 0.907 |
| 5 | -0.184 | 1.148 |
| 6 | 0.154 | 0.942 |
| 7 | 0.015 | 0.967 |
DATA1220-55, Fall 2024
2024-10-30
| Measure | Sample Statistic | Population Parameter |
|---|---|---|
| Mean | \(\bar{x}\) | \(\mu\) |
| Proportion | \(\hat{p}\) | \(p\) |
| Difference in Means | \(\bar{x}_1-\bar{x}_2\) | \(\mu_1-\mu_2\) |
| Difference in Proportions | \(\hat{p}_1 - \hat{p}_2\) | \(p_1 - p_2\) |
| Standard Deviation | \(s\) | \(\sigma\) |
Sufficient sample size
\(n \ge 30\) for \(\bar{x}\) (means)
\(n \ge 20\), \(n_{x=1} \ge 10\), & \(n_{x=0} \ge 10\) for \(\hat{p}\) (proportions)
The distribution of the sample statistic \(\bar{x}\) or \(\hat{p}\) approximates the normal distribution \(N\left(\text{population parameter}, \text{standard error}\right)\) as \(n \to \infty\).
\(\bar{x} \sim N\left(\mu, \frac{\sigma}{n}\right)\))
\(\hat{p} \sim N\left(p, \sqrt{\frac{p (1-p)}{n}}\right)\)
The sampling distribution is normal with \(\mu=\text{sample statistic}\) and \(\sigma=\operatorname{standard error}\).
The standard deviation of the sampling distribution of \(\bar{x}\) is the population standard deviation \(\sigma\) divided by the square root of the size of the sample \(n\).
\[ \begin{aligned} SE_{\bar{x}} &= \frac{\sigma}{\sqrt{n}} \end{aligned} \]
Because we don’t have access to the “true” value of \(\sigma\), we substitute the observed standard deviation in the sample \(s\) for inference and hypothesis testing.
\[ \begin{aligned} SE_{\bar{x}} &= \frac{s}{\sqrt{n}} \end{aligned} \]
The population in this figure has the “true” parameters of mean \(\mu=0\) and standard deviation \(\sigma=1\).
The sampling distribution is the distribution of sample statistics \(\bar{x}\) from samples with size \(n\) taken from the population \(x \sim N(0, 1)\), were you to sample infinite times.
Observed values of \(x\) are more variable than observed values of \(\bar{x}\).
| Sample ID | x_bar | s |
|---|---|---|
| 1 | -0.047 | 0.981 |
| 2 | 0.178 | 0.835 |
| 3 | 0.024 | 0.870 |
| 4 | -0.094 | 0.907 |
| 5 | -0.184 | 1.148 |
| 6 | 0.154 | 0.942 |
| 7 | 0.015 | 0.967 |
The standard deviation of the sampling distribution of \(\hat{p}\) for sample size \(n\) is…
\[ \begin{aligned} SE_{\bar{x}} &= \sqrt{\frac{p(1-p)}{n}} \end{aligned} \]
Because we don’t have access to the “true” value of \(p\), we substitute the observed statistic in the sample \(\hat{p}\) for inference and hypothesis testing.
\[ \begin{aligned} SE_{\hat{p}} &= \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \end{aligned} \]
| Sample ID | p_hat | SE |
|---|---|---|
| 1 | 0.533 | 0.091 |
| 2 | 0.667 | 0.086 |
| 3 | 0.400 | 0.089 |
| 4 | 0.500 | 0.091 |
| 5 | 0.433 | 0.090 |
| 6 | 0.600 | 0.089 |
| 7 | 0.667 | 0.086 |
DATA1220-55 Fall 2024, Class 24 | Updated: 2024-10-30 | Canvas | Campuswire